How Does Transition Energy Depend on Molecule Count in Quantum Dots?

[liran avraham]

1. Homework Statement
show that tthe transition energy can depend on the number of molecules in the quantum dot according to:
E∝1/N^2/3

in a intro course to nanotechnology class the prof gave us this question and said that "its very easy"
but i wasnt able to prove this.
can someone give me some guidance/insight?
i thought that the fractal is maybe an error and asked him but he said that its not.

 

[jambaugh]

Well, "quantum" means energy is associated with frequency and hence wavelength. How would the number of molecules affect this?

 

[Cthugha]

The fractal is correct and this is indeed one (a pretty oversimplifying one) of the very basic calculations to get people started.

First, consider the quantum dot as a one-dimensional particle in a box problem with infinite barriers. How does the energy of the ground state of this system scale with the width of the box?

Next, you consider the quantum dot as such a box. Consider it as a large sphere (the QD) made out of several small spheres (the molecules) The inside of the large sphere is the box. How does the radius of the large sphere depend on the number of molecules? Then put both results together.